#### Vol. 11, No. 1, 2017

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Logarithmic good reduction, monodromy and the rational volume

### Arne Smeets

Vol. 11 (2017), No. 1, 213–233
##### Abstract

Let $R$ be a strictly local ring complete for a discrete valuation, with fraction field $K$ and residue field of characteristic $p>0$. Let $X$ be a smooth, proper variety over $K$. Nicaise conjectured that the rational volume of $X$ is equal to the trace of the tame monodromy operator on $\ell$-adic cohomology if $X$ is cohomologically tame. He proved this equality if $X$ is a curve. We study his conjecture from the point of view of logarithmic geometry, and prove it for a class of varieties in any dimension: those having logarithmic good reduction.

##### Keywords
étale cohomology, logarithmic geometry, monodromy, nearby cycles, rational points
##### Mathematical Subject Classification 2010
Primary: 14F20
Secondary: 11G25, 11S15
##### Milestones
Revised: 6 July 2016
Accepted: 10 August 2016
Published: 23 January 2017
##### Authors
 Arne Smeets Radboud Universiteit Nijmegen IMAPP Heyendaalseweg 135 6525 AJ Nijmegen Netherlands University of Leuven Departement Wiskunde Celestijnenlaan 200B 3001 Heverlee Belgium